Problem: All of the 3rd grade teachers and students from Oak went on a field trip to an archaeology museum. Tickets were $$6.50$ each for teachers and $$3.50$ each for students, and the group paid $$30.50$ in total. The next month, the same group visited a science museum where the tickets cost $$26.00$ each for teachers and $$10.00$ each for students, and the group paid $$102.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6.5x+3.5y = 30.5}$ ${26x+10y = 102}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-26x-14y = -122}$ ${26x+10y = 102}$ Add the top and bottom equations together. $ -4y = -20 $ $ y = \dfrac{-20}{-4}$ ${y = 5}$ Now that you know ${y = 5}$ , plug it back into $ {6.5x+3.5y = 30.5}$ to find $x$ ${6.5x + 3.5}{(5)}{= 30.5}$ $6.5x+17.5 = 30.5$ $6.5x = 13$ $x = \dfrac{13}{6.5}$ ${x = 2}$ You can also plug ${y = 5}$ into $ {26x+10y = 102}$ and get the same answer for $x$ ${26x + 10}{(5)}{= 102}$ ${x = 2}$ There were $2$ teachers and $5$ students on the field trips.